Research Papers: Contact Mechanics

An Analytical Model of Four-Point Contact Rolling Element Ball Bearings

[+] Author and Article Information
Jacob D. Halpin

Halpin Engineering, LLC,
Torrance, CA 90504
e-mail: jake@HalpinEngineeringLLC.com

Anh N. Tran

ATEC Corp.,
Cypress, CA 90630
e-mail: tran.atec@gmail.com

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received June 20, 2015; final manuscript received February 20, 2016; published online May 18, 2016. Assoc. Editor: Xiaolan Ai.

J. Tribol 138(3), 031404 (May 18, 2016) (13 pages) Paper No: TRIB-15-1217; doi: 10.1115/1.4033134 History: Received June 20, 2015; Revised February 20, 2016

The purpose of this work is to establish an analytical model and standard way to predict the performance characteristics of a four-point contact, or gothic arch type, rolling element ball bearing. Classical rolling element bearing theory, as developed by Jones, has been extended to include the complex kinematics of the four-point contact bearing; thereby providing complete elementwise attitude and internal load distribution of the bearing under operating conditions. Standard performance parameters, such as element contact stresses, contact angles, inner ring deflections, nonlinear stiffness's, torque, and L10 life, are solved explicitly via standard Newton–Raphson techniques. Race control theory is replaced with a minimum energy state theory to allow both spin and slip to occur at the ball-to-raceway contact. The developed four-point model was programed within the orbis software program. Various test cases are analyzed and key analytical results are compared with the Jones four-point contact ball bearing analysis program, the Wind Turbine Design Guideline, DG03, and traditional two-point (angular contact) analysis codes. Model results for the internal distribution of ball loads and contact angles match the Jones program extremely well for all cases considered. Some differences are found with the DG03 analysis methods, and it is found that modeling a four-point contact bearing by overlaying two opposed angular contact bearings can result in gross errors.

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Hamrock, B. J. , and Anderson, W. J. , 1972, “ Arched-Outer-Race Ball-Bearing Analysis Considering Centrifugal Forces,” NASA Report No. TN D-6765.
Nelias, D. , and Leblanc, A. , 2007, “ Ball Motion and Sliding Friction in a Four-Contact-Point Ball Bearing,” ASME J. Tribol., 129(4), pp. 801–808. [CrossRef]
Jones, A. B. , 1964, “ The Mathematical Theory of Rolling-Element Bearings,” Mechanical Design and Systems Handbook, McGraw-Hill, New York.
Leveille, A. R. , 1997, “ The Non-Reversible Nature of Ball Bearing Internal Geometry,” REBG International Bearing Symposium, Orlando, FL.
Jones, A. B. , 1946, “ Analysis of Stresses and Deflections,” Vol. 1, General Motors Corp., Bristol, CT.
Harris, T. A. , 2001, Rolling Bearing Analysis, 4th ed., Wiley, New York.
Chapman, J. J. , and Boness, R. J. , 1975, “ The Measurement and Analysis of Ball Motion in High Speed Deep Groove Ball Bearings,” ASME J. Lubr. Technol., 97(3), pp. 341–348. [CrossRef]
ANSI, 1990, “ Load Ratings and Fatigue Life for Ball Bearings,” American National Standards Institute, New York, Standard No. Std. 9-1990.
Harris, T. A. , Rumbarger, J. H. , and Butterfield, C. P. , 2009, “ Wind Turbine Design Guideline, DG03: Yaw and Pitch Rolling Bearing Life,” National Renewable Energy Laboratory, Golden, CO, Report No. NREL/TP-500-42362.
ISO, 2006, “ Static Load Ratings,” International Organization for Standardization, Geneva, Switzerland, Standard No. ISO-76.


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Fig. 3

Normalized internal clearance circle depicting geometrical relationship between internal play, resting and free contact angles, and the shim size

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Fig. 2

Resting angle and free contact angle

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Fig. 1

Conventional deep groove bearing with arching region identified and resulting double arched bearing

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Fig. 4

Coordinate system showing relation to inner ring displacements

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Fig. 5

Ball position definition relative to global coordinates

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Fig. 6

Position of ball center and raceway curvature centers at initial and final loaded positions

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Fig. 8

Internal load distributions for a bearing under pure radial loading with various arching (expressed by resting angle)

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Fig. 9

Torque due to slip and spin bearing B under pure radial loading with various arching (expressed by resting angle)

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Fig. 7

Four-point contact stiffness diagram

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Fig. 10

Comparison of 4PT model to 2PT analysis with gyroscopic moment effects, bearing A, 5000 lbf thrust load: (a) normal ball loads, (b) contact angles, and (c) mean Hertzian stress

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Fig. 11

Comparison of 4PT model to 2PT analysis with gyroscopic moment effects, bearing B, 10,000 lbf thrust load: (a) normal ball loads, (b) contact angles, and (c) mean Hertzian stress



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